Ontogeny of a Modeler
David L Smith
2023-08-25
Library \(\uparrow\) Hub || [Model BuildModel-Building.html)
.
Models and modeling are being taken up and used by malaria control programs. This has created a demand for people with a variety of skills.
We need train academic modelers and mentors.
We need to train people who use models in programs
Should every modeler need to develop new models?
Can we be more efficient about training people with useful skills?
We need to introduce modeling to malaria experts and program specialists who have no mathematical training:
What is modeling?
Why is it useful?
NOTES
I’ll focus on the case of malaria, but we could describe a similar process for learning how to develop and apply models of other disesaes.
Starting with a Ross-Macdonald model
I’ll delve into some topics related to mosquitoes and vector control, but I won’t spend a ton of time talking about human epidemiology (i.e., infection,immunity, disease, treatment,detection…)
I’m going to take a birds-eye view of the process:
I’m not going to teach you anything (at least, not on purpose…)
I’m going to describe what I would have taught you, and you can make mental notes comparing this to what you actually learned, …
The point of all this is to reflect on the whole process we went through to get to where we are today, and to have a conversation…
I want to avoid getting bogged down in the details, so we’ll walk through the slides.
If you want to follow up:
Contact me at
smitdave@gmail.com(or ask Sam for my e-mail)The code that generated this html presentation is available on
github, and I’m more than happy to share it.
First Day
,
On day #1 of a two-day set of lectures, I want to narrate the ontogeny of an academic. In other words, I want to trace the first stages of professional development:
We learn to build and analyze models;
We teach ourselves (or learn from others) a large set of skills required to function as an effective academic.
We start to drive our own research agendas…
Basics
.
The Novice
Studying Populations
The Ross-Macdonald Model
Mathematical Analysis
Basic Skills
Programming (languages)
IDE (interactive development environment)
Commercial Software
Graphing / Visualization
Complementary skills
Ross-Macdonald
.
Review
- Smith, D.L., Battle, K.E., Hay, S.I., Barker, C.M., Scott, T.W., and McKenzie, F.E. (2012). Ross, Macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens. PLoS Pathog. 8, e1002588.
Ross (5+ papers, 2 books)
- Ross, R. (1911). Some quantitative studies in epidemiology. Nature 87, 466–467.
Lotka (7+ papers)
Lotka, A.J. (1923a). Contribution to the Analysis of Malaria Epidemiology. I. General Part. Am. J. Hyg. 3, 1–37.
Lotka, A.J. (1923b). Contribution to the analysis of malaria epidemiology. II. General part (continued) comparison of two formulae given. Am. J. Hyg. 3, 38–54.
Lotka, A.J. (1923c). Contribution to the Analysis of Malaria Epidemiology. III. Numerical Part*. Am. J. Epidemiol. 3, 55–95.
Lotka, A.J. (1923d). Contribution to the analysis of malaria epidemiology. V. Summary. Am. J. Hyg. 3, 113–121.
Sharpe, F.R., and Lotka, A.J. (1923). Contribution to the analysis of malaria epidemiology. IV. Incubation lag. Am. J. Hyg. 3, 96–112.
Macdonald (12+ papers, 1 book)
Macdonald, G. (1950a). The analysis of malaria parasite rates in infants. Trop. Dis. Bull. 47, 915–938.
Macdonald, G. (1950b). The analysis of infection rates in diseases in which superinfection occurs. Trop. Dis. Bull. 47, 907–915.
Macdonald, G. (1951). Community aspects of immunity to malaria. Br. Med. Bull. 8, 33–36.
Macdonald, G. (1952a). The analysis of the sporozoite rate. Trop. Dis. Bull. 49, 569–586.
Macdonald, G. (1952b). The analysis of equilibrium in malaria. Trop. Dis. Bull. 49, 813–829.
Macdonald, G. (1953). The analysis of malaria epidemics. Trop. Dis. Bull. 50, 871–889.
Macdonald, G. (1955a). The measurement of malaria transmission. Proc. R. Soc. Med. 48, 295–302.
Macdonald, G. (1955b). A new approach to the epidemiology of malaria. Indian J. Malariol. 9, 261–270.
Macdonald, G. (1956a). Epidemiological basis of malaria control. Bull. World Health Organ. 15, 613–626.
Macdonald, G. (1956b). Theory of the eradication of malaria. Bull. World Health Organ. 15, 369–387.
Macdonald, G. (1957). The epidemiology and control of malaria (Oxford university press).
Macdonald, G. (1965). Eradication of Malaria. Public Health Rep. 80, 870–879.
Garrett-Jones (9 papers)
Garrett-Jones, C. (1964a). The human blood index of malaria vectors in relation to epidemiological assessment. Bull. World Health Organ. 30, 241–261.
Garrett-Jones, C. (1964b). Prognosis for interruption of malaria transmission through assessment of the mosquito’s vectorial capacity. Nature 204, 1173–1175.
Garrett-Jones, C., and Grab, B. (1964). The assessment of insecticidal impact on the malaria mosquito’s vectorial capacity, from data on the proportion of parous females. Bull. World Health Organ. 31, 71–86.
De Zulueta, J., and Garrett-Jones, C. (1965). An Investigation of the Persistence of Malaria Transmission in Mexico. Am. J. Trop. Med. Hyg. 14, 63–77.
Bruce-Chwatt, L.J., Garrett-Jones, C., and Weitz, B. (1966). Ten years’ study (1955-64) of host selection by anopheline mosquitos. Bull. World Health Organ. 35, 405–439.
Garrett-Jones, C., Dranga, A., Marinov, R., Mihai, M., Organization, W.H., and Others (1968). Epidemiological entomology and its application to malaria (Geneva: Geneva: World Health Organization).
Garrett-Jones, C., and Pal, R. (1969). Insecticide resistance in malaria vectors: general review of the” neotropical, palearctic and oriental regions. Cahiers ORSTOM. Série Entomologie Médicale et Parasitologie 7, 3–8.
Garrett-Jones, C., and Shidrawi, G.R. (1969). Malaria vectorial capacity of a population of Anopheles gambiae: an exercise in epidemiological entomology. Bull. World Health Organ. 40, 531–545.
Garrett-Jones, C. (1970). Problems of epidemiological entomology as applied to malariology. Entomol Soc Amer Misc Publ.
DDE
- Aron, J.L., and May, R.M. (1982). The population dynamics of malaria. In The Population Dynamics of Infectious Diseases: Theory and Applications, R.M. Anderson, ed. (Boston, MA: Springer US), pp. 139–179.
System of Delay Differential Equations
\[\begin{equation} \begin{array}{rl} \frac{dX}{dt} & = f q \; \frac{b Z}{H} (H-X) - r X \\ \frac{dY}{dt} & = f q \; \frac{cX}{H} (M-Y) - g Y \\ \frac{dZ}{dt} & = f q \; \frac{c X_{t-\nu}}{H} e^{-g\nu} (M-Y_{t-\nu}) - g Z \\ \end{array}\end{equation}\]
Variables
\(X\) - density of infected humans
\(Y\) - density of infected mosquitoes
\(Z\) - density of infective mosquitoes
Parameters
\(H\) - density of humans
\(M\) - density of mosquitoes
\(\nu\) - EIP (in days)
\(g\) - daily, per-capita mosquito death rate (lifespan is \(1/g\))
\(f\) - daily, per-capita mosquito blood feeding rate
\(q\) - human fraction (human blood meals / all blood meals)
\(r\) - daily, per-capita clearance rate of parasite infections (duration is \(1/r\))
\(b\) - fraction of infective bites causing an infection in humans
\(c\) - fraction of bites on infected humans infecting a mosquito
ODE
- Aron, J.L., and May, R.M. (1982). The population dynamics of malaria. In The Population Dynamics of Infectious Diseases: Theory and Applications, R.M. Anderson, ed. (Boston, MA: Springer US), pp. 139–179.
System of Differential Equations
\[\begin{equation} \begin{array}{rl} \frac{dX}{dt} & = f q \; e^{-g \nu} \; \frac{b Y}{H} (H-X) - r X \\ \frac{dY}{dt} & = f q \; \frac{cX}{H} (M-Y) - g Y \\ \end{array}\end{equation}\]
Variables
\(X\) - density of infected humans
\(Y\) - density of infected mosquitoes
Term (Implied)
\(Z\) - density of infective mosquitoes, \(Z=e^{-g\nu} Y\)
Parameters
\(H\) - density of humans
\(M\) - density of mosquitoes
\(\nu\) - EIP (in days)
\(g\) - daily, per-capita mosquito death rate (lifespan is \(1/g\))
\(f\) - daily, per-capita mosquito blood feeding rate
\(q\) - human fraction (human blood meals / all blood meals)
\(r\) - daily, per-capita clearance rate of parasite infections (duration is \(1/r\))
\(b\) - fraction of infective bites causing an infection in humans
\(c\) - fraction of bites on infected humans infecting a mosquito
DE
System of Difference Equations
\[\begin{equation} \begin{array}{rl} X_{t+1} &= e^{-r} X_t + \left[1-e^{-f q \; e^{-g \nu} \; \frac{b Y}{H}} \right] (H-X_t) \\ Y_{t+1} & = e^{-g} Y_t + \left[1-e^{-f q \; \frac{cX}{H}} \right] (M-Y_t) \\ \end{array}\end{equation}\]
Variables
\(X_t\) - density of infected humans on day \(t\)
\(Y_t\) - density of infective mosquitoes on day \(t\)
Term (Implied)
\(Z_t\) - density of infective mosquitoes, \(Z_t=e^{-g\nu} Y_t\)
Parameters
\(H\) - density of humans
\(M\) - density of mosquitoes
\(\nu\) - EIP (in days)
\(g\) - daily, per-capita mosquito death rate (lifespan is \(1/g\))
\(f\) - daily, per-capita mosquito blood feeding rate
\(q\) - human fraction (human blood meals / all blood meals)
\(r\) - daily, per-capita clearance rate of parasite infections (duration is \(1/r\))
\(b\) - fraction of infective bites causing an infection in humans
\(c\) - fraction of bites on infected humans infecting a mosquito
sDE
System of Stochastic Difference Equations
\[\begin{equation} \begin{array}{rl} X_{t+1} =& \mbox{binom}\left(X_t, e^{-r} + \left(1-e^{-r}\right) \left[1-e^{-f q \; e^{-g \nu} \; \frac{b Y}{H}} \right] \right) \\ &+ \mbox{binom}\left(H-X_t, 1-e^{-f q \; e^{-g \nu} \; \frac{b Y_t}{H}} \right) \\ Y_{t+1} =& \mbox{binom}\left(Y_t, e^{-g} \right) + \mbox{binom}\left(M-Y_t, 1-e^{-f q \; \frac{cX_t}{H}} \right) \\ \end{array}\end{equation}\]
Variables
\(X_t\) - density of infected humans on day \(t\)
\(Y_t\) - density of infective mosquitoes on day \(t\)
Term (Implied)
\(Z_t\) - density of infective mosquitoes, \(Z_t=e^{-g\nu} Y_t\)
Parameters
\(H\) - density of humans
\(M\) - density of mosquitoes
\(\nu\) - EIP (in days)
\(g\) - daily, per-capita mosquito death rate (lifespan is \(1/g\))
\(f\) - daily, per-capita mosquito blood feeding rate
\(q\) - human fraction (human blood meals / all blood meals)
\(r\) - daily, per-capita clearance rate of parasite infections (duration is \(1/r\))
\(b\) - fraction of infective bites causing an infection in humans
\(c\) - fraction of bites on infected humans infecting a mosquito
sODE
The Gillespie Algorithm
Motivated by this set of equations:
\[\begin{equation} \begin{array}{rl} \frac{dX}{dt} & = f q \; e^{-g \nu} \; \frac{b Y}{H} (H-X) - r X \\ \frac{dY}{dt} & = f q \; \frac{cX}{H} (M-Y) - g Y \\ \end{array}\end{equation}\]
- Let \(t_i\), \(i=0,1,2,...\) index the time when a set of events occurred
Variables
\(X_i\) - number of infected humans at time \(t_i\)
\(Y_i\) - number of infected mosquitoes at time \(t_i\)
Term (Implied)
\(Z_i\) - density of infective mosquitoes, \(Z_i=e^{-g\nu} Y_i\)
Pseudo-Code
Set the initial state
Do one event:
Compute the total rate of change in the system: \[\lambda = fqe^{g\nu}\frac{bY_i}{H}(H-X_i) + r X_i + fq \frac{cX_i}{H}(M-Y_i) + g Y_i\]
Compute the time to the next event: \[t_{i+1} = t_{i} + \mbox{rexp}(\lambda))\]
Determine which event:
Human Infected: \(X_{i+1} = X_i + 1\) with probability \(fqe^{g\nu}\frac{bY_i}{H}(H-X_i)/\lambda\)
Human Infection Cleared: \(X_{i+1} = X_i - 1\) with probability \(rX_i/\lambda\)
Mosquito Infected: \(Y_{i+1} = Y_i + 1\) with probability \(fq \frac{cX_i}{H}(M-Y_i)/\lambda\)
Mosquito Died: \(Y_{i+1} = Y_i - 1\) with probability \(g Y_i/\lambda\)
Update and repeat
Parameters
\(H\) - density of humans
\(M\) - density of mosquitoes
\(\nu\) - EIP (in days)
\(g\) - daily, per-capita mosquito death rate (lifespan is \(1/g\))
\(f\) - daily, per-capita mosquito blood feeding rate
\(q\) - human fraction (human blood meals / all blood meals)
\(r\) - daily, per-capita clearance rate of parasite infections (duration is \(1/r\))
\(b\) - fraction of infective bites causing an infection in humans
\(c\) - fraction of bites on infected humans infecting a mosquito
Math
Orbits
require(deSolve)## Loading required package: deSolveRM_derivs = function(t,y,p){with(as.list(c(y,p)),{
dX = f*q*exp(-g*eip) *b*Y/H*(H-X) - r*X
dY = f*q*c*X/H*(M-Y) - g*Y
list(c(dX,dY))
})}params = c(f=.3, q=.9, g=1/12, eip=10,b=.55, r=1/200, H=100,c=.1, M=200)
inits = c(X=1, Y=0)
tt = 0:500
orbits = data.frame(ode(inits,tt,RM_derivs,params)) with(as.list(params),{
plot(tt, orbits$X/H, type = "l", col = "darkred", xlab = "Time (Days)", ylab = "x,z")
lines(tt, orbits$Y*exp(-g*eip)/M, col = "darkblue")
})
Steady States
Set derivatives equal to zero
\[\begin{equation} \begin{array}{rl} 0 = \frac{dX}{dt} & = f q \; e^{-g \nu} \; \frac{b Y}{H} (H-X) - r X \\ 0 = \frac{dY}{dt} & = f q \; \frac{cX}{H} (M-Y) - g Y \\ \end{array}\end{equation}\]
Change variables
Let \(x=X/H\) and \(y=Y/M\) and \(m=M/H\)
\[\begin{equation} \begin{array}{rl} 0 & = f q \; e^{-g \nu} \; b y m (1-x) - r x \\ 0 & = f q \; cx (1-y) - g y \\ \end{array}\end{equation}\]
Solve for \(y\)
\[\begin{equation} y = \frac{f q c x} {g + fqcx} \end{equation}\]
Substitute \(y\)
\[\begin{equation} 0 = f q \; e^{-g \nu} \; b m \frac{f q c x} {g + fqcx} (1-x) - r x \end{equation}\]
we pull out \(x\):
\[\begin{equation} 0 = x \left( f q \; e^{-g \nu} \; b m \frac{f q c} {g + fqcx} (1-x) - r \right) \end{equation}\]
Obviously \(x=0\) is a steady state, so we rearrange the other part to find the other steady state:
\[\begin{equation} f q \; e^{-g \nu} \; b m \frac{f q c} {g + fqcx} (1-x) = r \end{equation}\]
or
\[\begin{equation} f^2 q^2 \; e^{-g \nu} \; bc m (1-x) = r (g+fcx) \end{equation}\]
Now let \[R_0 = \frac{f^2 q^2 \; e^{-g \nu} bc m}{gr}\]
and \[s = fq/g\]
I can rewrite this as:
\[\begin{equation} R_0 (1-x) = 1+csx \end{equation}\]
so
\[\begin{equation} x = \frac{R_0-1}{R_0 + cs} \end{equation}\]
cs=1
R0 = seq(0.8, 4, length.out=200)
plot(R0, (R0-1)/(R0+cs), type = "l", ylab = expression(x), xlab= expression(R[0]))
segments(0.8,0,4,0)
Stability
Notation
\[\begin{equation} \begin{array}{rl} \frac{dX}{dt} & = f q \; e^{-g \nu} \; \frac{b Y}{H} (H-X) - r X = F(X,Y) \\ \frac{dY}{dt} & = f q \; \frac{cX}{H} (M-Y) - g Y = G(X,Y)\\ \end{array}\end{equation}\]
Let \(\bar X, \bar Y\) denote a steady state, so \(F(\bar X,\bar Y) = G(\bar X, \bar Y) = 0\).
Motivation
What happens to orbits that start at \(\bar X + \epsilon, \bar Y + \epsilon\)?
Do they approach \(\bar X, \bar Y\) or do they get further away?
If orbits return to the equilibrium, we say that the equilibrium is stable.
If orbits get further from the equilibrium, we say that the equilibrium is unstable.
Linearize
We apply Taylor’s theorem. We linearize to get a new linear system of equations that has the same local behavior as the original system:
\[\begin{equation} \left[ \begin{array}{c} \frac{dX}{dt} \\ \frac{dY}{dt} \end{array} \right] = \left[ \begin{array}{rl} \frac{\partial F}{dX} & \frac{\partial F} {dY} \\ \frac{\partial G}{dX} & \frac{\partial G} {dY} \\ \end{array} \right] \left[ \begin{array}{c} X \\ Y \end{array} \right] \end{equation}\]
The solutions to these linear systems with eigenvalues \(\lambda_i\) and associated eigenvectors \(E_i\) all look like this:
\[\begin{equation} \left[\begin{array}{c} X(t) \\ Y(t) \end{array}\right] = c_1 E_1 e^{\lambda_1 t} + c_2 E_2 e^{\lambda_2 t} \end{equation}\]
so to answer our question, all we really need to know is if the largest eigenvalue is positive.
\[\begin{equation} \mbox{eigs} \left( \left. \left[ \begin{array}{rl} \frac{\partial F}{dX} & \frac{\partial F} {dY} \\ \frac{\partial G}{dX} & \frac{\partial G} {dY} \\ \end{array} \right] \right|_{\bar X, \bar Y} \right) \end{equation}\]
Do it
\[\begin{equation} \left[ \begin{array}{rl} - f q \; e^{-g \nu} \; \frac{b Y}{H} - r & f q \; e^{-g \nu} \; \frac{b }{H} (H-X) \\ f q \; \frac{c}{H} (M-Y) & - f q \; \frac{cX}{H} - g \\ \end{array} \right] \end{equation}\]
Is (0,0) stable?
At \((\bar X, \bar Y) = (0,0)\)
\[\begin{equation} \left[ \begin{array}{rl} - r & b f q \; e^{-g \nu} \\ f q \; \frac{cM}{H} & - g \\ \end{array} \right] \end{equation}\]
The eigenvalues are solutions to \[\lambda^2 - (r+g) \lambda + rg (1-R_0)\] or \[ \lambda = \frac{(r+g) \pm \sqrt{(r+g)^2 -4 rg (1-R_0)}}{2}\] By inspection, if \(R_0>1\) then \(\max{\lambda}>0.\)
Is X,Y stable?
At \((\bar X, \bar Y) > (0,0)\)
\[\begin{equation} \left[ \begin{array}{rl} - \frac{rH}{H-\bar X} & -r \frac{\bar X}{\bar Y} \\ -g \frac{\bar Y}{\bar X} & - \frac{gM}{M-\bar Y} \\ \end{array} \right] \end{equation}\]
The characterisitic equation is \[\lambda^2 + \left(\frac{rH}{H-\bar X} + \frac{gM} {M-\bar Y}\right) \lambda + \frac{rH}{H-\bar X}\frac{gM} {M-\bar Y}- rg\] or \[\lambda^2 + \left(\frac{rH}{H-\bar X} + \frac{gM} {M-\bar Y}\right) \lambda + rg \left(\frac{M} {M-\bar Y} \frac{H}{H-\bar X}- 1 \right)\] The eigevalues are: \[ \frac{-\left(\frac{rH}{H-\bar X} + \frac{gM} {M-\bar Y}\right) \pm \sqrt{ \left(\frac{rH}{H-\bar X} + \frac{gM} {M-\bar Y}\right)^2- 4 rg \left(\frac{M} {M-\bar Y} \frac{H}{H-\bar X}- 1 \right) } }{2}\] But \[ \frac{M} {M-\bar Y} \frac{H}{H-\bar X}> 1\] so by inspection, both eigenvalues are negative.
If \(R_0 > 1\) and the equilibrium exists, then it is stable.
Advanced…
We would go back and solve for the steady states in the DDE model
We would do the stability analysis for the DE model
We would discuss the problem of having an absorbing state in stochastic models.
We would construct a two-patch model
We would compute the next generation matrix
We would discuss threshold criteria for heterogeneous dynamics:
We would introduce the spectral average
We would compute threshold conditions for models with heterogeneity
We would construct a model with temporally forced parameters:
We would discuss the concept of a canonical seasonal signal
We would discuss the concept of stable orbits
We would compute the temporal eigenvector and threshold conditions
Vector Control
.
Macdonald
Macdonald framed vector control as an activity that modified transmission. The effect of vector control could be understood through changes in the bionomic parameters.
In the 1952 paper, Macdonald showed that the sporozoite rate was highly sensitive to daily mosquito survival.
Elasticity
Extended VC
Policy goals generally involve reducing transmission down to some target level. In the case of elimination, this requires reaching an effect size sufficient to reduce \(R_C < 1\) (i.e., above the dotted line in A–C). Under certain situations this cannot be achieved through scaling-up coverage of a single intervention alone, including: (3A) high baseline transmission (insecticide treated bednets [ITNs] and larval source management [LSM]); (3B) multiple vector species (red and black lines denote a setting where half of vectorial capacity (VC) is due to a species that is insecticide resistant [IR] but still susceptible to LSM in comparison to the blue line where all species are susceptible to all interventions); (3C) mosquito biting plasticity reduces the effectiveness of ITNs (in the red line feeding frequency is unaffected in mosquitoes with opportunistic biting patterns due to the availability of non-human hosts); (3D) the spread of insecticide resistance (plots show the change in effect size as ITN coverage is scaled up to 80% [grey shaded bar] then resistance emerges at half the rate of ITN scale up [fast, red line] or one tenth the rate of scale up [slow, blue line]). Dotted lines show the effect of a second ITN campaign where nets are replaced with a different insecticide.
Effect Sizes vs. Coverage
How are the bionomic parameters modified by interventions?
The probability of contact is a function of coverage
The entire feeding cycle is modified
Basic Skills
.
Math Software (Mathematica, Maple, Matlab,…)
Stats Software (Stata, SPSS, …)
Programming
Low Level: C++, Fortran, Basic, Java, …
High Level: R, Julia, Python, …
git
IDE
Interactive Development Environment
R-Studio
Jupyter Notebooks
Graphing / Visualization
Complementary Skills
Over time, we encounter new ideas and skills…
Verification
- Kolmogorov forward equations
Data / Simulation
Probability distribution functions
If the variance of the data is roughly equal to the mean, then use a Poisson.
If the varieance is larger than the mean, then use a Negative Binomial distribution.
Mixture distributions
The negative binomial can be derived as a mixture distribution where the expected value is Gamma distributed, and the variate is Poisson-distributed, for a given mean.
The beta-binomial can be derived as a mixture distribution where the expected value of a proportion has a distribution, but the variate is then Boolean
Sub-sampling (bootstrap, jacknife)
- The funcion
sampleinRallows you to subsample a dataset
- The funcion
Why?
.
Why model malaria?
- To understand how something works
- To develop theory (Hardy-Weinberg)
- To frame a problem (\(R_0\) / thresholds)
- To describe something accurately (linear regression)
- To predict (weather report)
- To publish papers
- To analyze data
- To support a decion / advise on a policy
- For inference (A causes B)
- Power analysis (RCT)
- Estimation (How long will it take?)
- Persuation, intimidation, ….
Adequacy…
Are the models adequate to this tasks?
Frameworks
.
A model is an object that produces orbits.
A model family is a mathematical object that fully defines a set of inter-related models that may differ from one another:
parameter values
initial conditions
random number seed (for stochastic models)
A model framework is a way of constructing inter-related model families.
Modularity
.
Decouple
\[\begin{equation} \begin{array}{rl} \frac{dX}{dt} & = f q \; e^{-g \nu} \; \frac{b Y}{H} (H-X) - r X \\ \frac{dY}{dt} & = f q \; \frac{cX}{H} (M-Y) - g Y \\ \end{array}\end{equation}\]
Rewrite as:
\[\begin{equation} \begin{array}{rl} \frac{dX}{dt} & = h (H-X) - r X \\ \frac{dY}{dt} & = f q \kappa (M-Y) - g Y \\ \end{array}\end{equation}\]
Connecting Terms
\[\begin{equation} \begin{array}{rl} h & = f q \; e^{-g \nu} \; \frac{b Y}{H} \\ \kappa & = \frac{cX}{H} \\ \end{array}\end{equation}\]
Extend
Dynamics
\[\begin{equation} \begin{array}{rl} \frac{dL}{dt} & = \nu - (\phi + \theta + \sigma L) L \\ \frac{dM}{dt} & = \Lambda - g M \\ \frac{dY}{dt} & = f q \kappa (M-Y) - g Y \\ \frac{dX}{dt} & = h (H-X) - r X \\ \frac{dH}{dt} & = f_H (H, \ldots)\\ \end{array}\end{equation}\]
Connecting Terms
\[\begin{equation} \begin{array}{rl} \nu & = f \omicron M \\ \Lambda & = \phi L \\ \kappa & = \frac{cX}{H} \\ h & = f q \; e^{-g \nu} \; \frac{b Y}{H} \\ \end{array}\end{equation}\]
Generalize
Dynamics
\[\begin{equation} \begin{array}{rl} \frac{d {\cal L}}{dt} & = \nu - F_{\cal L}({\cal L}, ...) \\ \frac{d {\cal M}}{dt} & = \Lambda - F_{\cal M}({\cal M}, ...) \\ \frac{d {\cal Z}}{dt} & = F_{\cal Z}\left(\kappa, {\cal M}, {\cal Z}, ... \right) \\ \frac{d {\cal X}}{dt} & = F_{\cal X}\left(h, {\cal X}, {\cal H}, ... \right) \\ \frac{d {\cal H}}{dt} & = F_{\cal H}\left({\cal H}, {\cal X}, \ldots \right) \\ \end{array}\end{equation}\]
Connecting Terms
\[\begin{equation} \begin{array}{rl} \nu & = F_\nu({\cal M}) \\ \Lambda & = F_\Lambda({\cal L}) \\ \kappa & = F_\kappa ({\cal X}) \\ h & = F_h ({\cal Z}) \\ \end{array}\end{equation}\]
Availability
What are the constraints on \(q\), the human fraction?
If there are no human blood hosts, then \(q=0\)
If there are no other hosts, then \(q=1\)
What are the constraints on \(f\), the blood feeding rate?
If there are no hosts, then \(f=0\).
If there are lots of hosts, but they are all protected by nets, then blood feeding should be slower.
Availability
We stratify the human population into sub-populations, according to their availability to mosquitoes. The density of humans in each stratum is \(H_i\), and we assign each stratum a biting weight, \(w_i\). Total human availability is \[ W= \sum_i w_i H_i.\] Similarly, we let \(O\) represent the availability of all other potential vertebrate blood hosts, and we let \(\zeta\) be a scaling factor such that: \[B = W + O^\zeta.\] We let \[q= \frac{B}{W}\] and we let \[f = f_x \frac{s_f B} {1 + s_f B}\]
par(mfrow = c(2,1))
fx = .5
sf = .1
B = seq(0, 50, length.out=250)
ff = fx*sf*B/(1+sf*B)
plot(B, ff, type = "l", xlab = "Availability", ylab = "Feeding rate")
plot(B, 1/ff, type = "l", xlab = "Availability", ylab = "Time to Feedn (Days)", ylim = c(0, 5))
Spatial Dynamics
.
To create a generalized framework for heterogeneous spatial transmission dynamics, we need to be able to:
Subdivide the landscape into patches
Segment heterogeneous human populations into strata
Model the spatial dynamics of immature mosquito populations in a set of independent aquatic habitats
To make all of this work, we need to worry about availability in a dynamic host population, so we have completely recongfigured the model for blood feeding.
Spatial Blood Feeding
Micro-simulation
##### Macro
Elaboration
.
By topic:
Malaria Epidemiology, including infection, immunity, disease, treatment and chemprotection, infectiousness, and detection.
Human Behavior, including care seeking, travel, mobility, and ITN usage.
Adult Mosquito
Behavior mating, blood feeding, egg laying, sugar feeding, search
Dispersal How / why do mosquitoes move around?
Weather
Aquatic Mosquito
Demographics density dependent and density independent mortality, maturation,
Biotic Factors Resources, competitors, predators,…
Weather Habitat dynamics modified by rainfall, temperature
Adult Mosquito
.
State Space
Infection States (infected, infectious)
Behavioral States (mating, blood feeding, egg laying, sugar feeding, resting,…)
Reproductive States (mated, parous, gravid,…)
Energetic States (fed, starving,…)
Damaged States (insecticide exposed,…)
Active / Quiescent / Estevating
State Space
.
MBITES
Mosquito Movement
Movement & Dispersal
Diel activity
Indoor vs. outdoor
Resource availability
Search and availability
Repellancy
Aquatic Ecology
Density dependence
Delayed maturation leads to non-monotonic relationships between eggs laid and adults emerging
The effects are exacerbated by stage-structure
Weather & Hydrology: Habitat dynamics, including flushing, dessication, …
Larval Source Management
Wrap
Complexity
- Individual-Based Simulation Models
What are the major missing / underdeveloped elements?
Theory for vector control
Mosquito ecology
Sampling theory
Evolution of insecticide resistance,
…
Should Everyone Learn Everything?
Can you compute the \(\sqrt{2}\)? Do you need to?
Can we identify training tracks?
Math (Academic)
Computation (Academic)
Basic Research (Academic)
Policy
How can we team up better?
Second Day
.
On day #2 of a two-day set of lectures, I want to focus on the way we use models in science and policy.
Inference
.
- Population sciences are different…
Ross, 1916
It is somewhat suprising that so little mathematical work should have been done on the subjct of epidemics, and, indeed, on the distribution of diseases in general. Not only is the theme of immediate importance to humanity, but it is one which is fundamentally connected with numbers, while vast masses of statistics have long been awaiting proper examination. But, more than this, many and indeed the principle problems of epidemiology on which preventive measures largely depend, such as the rate of infection, the frequency of outbreaks, and the loss of immunity, can scarcely ever be resolved by any other methods than those of analysis. For example, infectious diseases may perhaps be classified in three groups: (1) diseases such as leprosy, tuberculosis, and (?) cancer, which fluctuate comparatively little from month to month, though they may slowly increase or decrease over the course of years; (2) diseases such as measles, scarlatina, malaria, and dysentery, which though constantly present in many countries, flare up in epidemics and frequent intervals; and (3) disease such as plague or cholera, whicch disappear entirely after periods of acute epidemicity.
To what are these differences due? Why indeed should epidemics occur at all, and why should not all infectious diseases belong to the first group and always remain at a flat rate? Behind these phenomena must be causes which are of profound importance to mankind and which probably can be ascertained only by those principles of careful computation which have yielded such brilliant results in astronomy, physic, and mechanics….
Ross, 1916
The whole subject is capable of study by two distinct methods which are used in other branches of science, which are complementary of each other, and which should always converge to the same results – the a posteriori and the a priori methods. In the former we commence with observed statistics, endeavour to fit analytical laws to them, and so work backwards to the underlying cause (as done in much statistical work of the day); and in the latter we assume a knowledge of the causes, construct our differential equations on that supposition, follow up on the logical consequences, and finally test the calculated results by comparing them with the observed statistics.
Closing the Loop
.
Three longitudinal studies were conducted in Kihihi, Kanungu; Walakuba, Jinja; and Nagongera, Tororo, Uganda.
We wanted to estimate a household biting weight for each one of the ~100 households enrolled in the study using catch counts from CDC-LTs.
What methods would give accurate estimates?
Kang SY, et al. (2018). Heterogeneous exposure and hotspots for malaria vectors at three study sites in Uganda. Gates Open Res 2, 32.
Simulation Study
Generate a signal with a known seasonal pattern, \(S_d\).
Assign each household a biting weight, \(w_h\).
Assume that the counts were negative binomial with the given mean, \[M \sim \mbox{rnbinom}\left(S_d, w_h, sz \right)\]
Sample the houses approximately once per month, letting the day of the month vary.
Use various statistical models to estimate \(S_d\) and \(w_h\).
****
We would have come very close to getting the right answers in many cases if we had taken the normalized mosquito densities.
We went on to investigate whether these rates varied in the wet vs. dry season, whether we could detect “hostpots” and whether there were any differences in transmission, comparing the first half to the second half of the season.
Parsimony / Model Selection
Parsimony Model selection is more or less based on the trade-off between bias and variance versus the number of estimable parameters in the model. The principle of parsimony tells us that as we increase the number of parameters in a model the bias decreases but the variance increases.
Three Model
Selection Criteria:
- Akaike’s Information Criteria
- Bayesian Information Criteria
- Deviance Information Criteria
State Space Modeling
.
The true state of the system is represented in a model
The observable states are modeled with an explicit sampling process.
Fig 1
Fig 2
Adaptive Management
Big Picture
Goals – The goals of malaria control are to minimize the health burden of malaria and, ultimately, to eliminate malaria in Uganda, in alignment with a global, long-term goal to eradicate malaria
Policy – Effective policy making requires a thorough understanding of the current malaria situation and the tradeoffs between interventions. Tradeoffs arise because of a budget constraint and uncertainty about the true effects of each intervention in different contexts. We want to stratify Uganda and optimize resource allocation to save lives and quickly eliminate malaria.
Advocacy – For purposes of advocacy, we would also like to estimate the burden of malaria, the averted burden, and the cost and timeline of malaria elimination. Further, we want to be able to predict how these measures would change if the budget were increased.
Details
Analytics: Navigating tradeoffs requires a working knowledge of malaria epidemiology, malaria transmission, health systems, human behaviors, vector control, and costs. With so many moving parts, we need sophisticated analytic tools to construct an interpretable synthesis.
Uncertainty: With imperfect information and a constantly changing body of evidence, we will also need robust algorithms for dealing with uncertainty.
Information Systems: We need streamlined information systems to reduce the human workloads required to generate timely insights during repeated reporting on key malaria metrics.
Collaboration: We are proposing a collaboration between the Uganda Ministry of Health, the University of Washington, and an analytics team in Uganda.
Adaptive Malaria Control: Working together, we will develop protocols to review malaria policy, update surveillance systems, and adapt to changing conditions.
Metrics
Transmission
- Prevalence: in older children (aged 5-10), by RDT, period average
- Infection Incidence: in older children, # infections/person/yr, period average
- annual Entomological Inoculation Rate: # infective bites/person/yr, CDC-LT
- Malaria Incidence: young children (<5), # clinical episodes/person/yr, period average
Care Seeking
- Drug Taking: ddd/person /yr, period average
- Compliance: % adhering to full drug regimen
- Care Seeking: not / public / private OR prompt / delayed
- Drug Resistance: % 28-day parasitological failure
Burden
- Severe Malaria Incidence: # clinical episodes/person/yr, period average
Vector Control Coverage
- IRS: % of houses sprayed. When did spraying occur?
- ITN: % sleeping under a net
Analytics
.
Science vs. Policy
Uncertainty
Robust Analytics for Malaria Policy
.
Estimation Framework
Robustness
RAMP
Pseudocode
SBA
Identify a set of plausible scenarios
Pick a model
Fit it to data
Contruct a counterfactual
Develop a forecast
Compare Scenarios
Recommend a Policy:
Compare Analyses
Replicate SBA, store inputs and recommendations
Compare policies
If there’s no consensus, examine the ensemble of recommendations to identify discriminating factors – what unkown factors accounted for the most relevant differences in recommended policies?
Make a policy recommendation
Pass on a prioritized list of policy recommendations
Adaptive Malaria Control
MicroMoB: Discrete Time Simulation of Mosquito-Borne Pathogen Transmission. Micro-MoB is a modular framework for discrete time simulation for simulating mosquito-borne pathogen transmission dynamics and control.
xDE-MoB is a planned satellite package to facilitate construction and analysis of systems of differential equations.
Other satellite